Fourier Analysis D. Gordon E. Robertson, PhD, FCSB School of Human Kinetics University of Ottawa

7/3/2015 Biomechanics Laborartory, University of Ottawa2   In theory: Every periodic signal can be represented by a series (sometimes an infinite series) of sine waves of appropriate amplitude and frequency.   In practice: Any signal can be represented by a series of sine waves.   The series is called a Fourier series.   The process of converting a signal to its Fourier series is called a Fourier Transformation. Why use Fourier Analysis?

7/3/2015 Biomechanics Laborartory, University of Ottawa3 Generalized Equation of a Sinusoidal Waveform   w(t) = a 0 + a 1 sin (2  f t +  )   w(t) is the value of the waveform at time t

7/3/2015 Biomechanics Laborartory, University of Ottawa4 Generalized Equation of a Sinusoidal Waveform   w(t) = a 0 + a 1 sin (2  f t +  )   a 0 is an offset in units of the signal   Offset (also called DC level or DC bias): mean value of the signal AC signals, such as the line voltage of an electrical outlet, have means of zero

7/3/2015 Biomechanics Laborartory, University of Ottawa5 Offset Changes

7/3/2015 Biomechanics Laborartory, University of Ottawa6 Generalized Equation of a Sinusoidal Waveform   w(t) = a 0 + a 1 sin (2  f t +  )   a 1 is an amplitude in units of the signal   Amplitude: difference between mean value and peak value sometimes reported as a peak-to-peak value (i.e., a p-p = 2 a)

7/3/2015 Biomechanics Laborartory, University of Ottawa7 Amplitude Changes

7/3/2015 Biomechanics Laborartory, University of Ottawa8 Generalized Equation of a Sinusoidal Waveform   w(t) = a 0 + a 1 sin (2  f t +  )   f is the frequency in cycles per second or hertz (Hz)   Frequency: number of cycles (n) per second sometimes reported in radians per second   (i.e., w = 2  f ) can be computed from duration of the cycle or period (T): (f = n/T)

7/3/2015 Biomechanics Laborartory, University of Ottawa9 Frequency Changes

7/3/2015 Biomechanics Laborartory, University of Ottawa10 Generalized Equation of a Sinusoidal Waveform   w(t) = a 0 + a 1 sin (2  f t +  )    is phase angle in radians   Phase angle: delay or phase shift of the signal can also be reported as a time delay in seconds e.g., if , sine wave becomes a cosine

7/3/2015 Biomechanics Laborartory, University of Ottawa11 Phase Changes

7/3/2015 Biomechanics Laborartory, University of Ottawa12 Generalized Equation of a Fourier Series   w(t) = a 0 +  a i sin (2  f i t + q i )   since frequencies are measured in cycles per second and a cycle is equal to 2  radians, the frequency in radians per second, called the angular frequency, is:  = 2  f   therefore: w(t) = a 0 +  a i sin (  i t + q i )

7/3/2015 Biomechanics Laborartory, University of Ottawa13 Alternate Form of Fourier Transform   an alternate representation of a Fourier series uses sine and cosine functions and harmonics (multiples) of the fundamental frequency   the fundamental frequency is equal to the inverse of the period (T, duration of the signal): f 1 = 1/period = 1/T   phase angle is replaced by a cosine function   maximum number in series is half the number of data points (number samples/2)

7/3/2015 Biomechanics Laborartory, University of Ottawa14 Fourier Coefficients   w(t) = a 0 +  [ b i sin (  i t) + c i cos (  i t) ]   b i and c i, called the Fourier coefficients, are the amplitudes of the paired series of sine and cosine waves (i=1 to n/2); a 0 is the DC offset   various processes compute these coefficients, such as the Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT)   FFTs compute faster but require that the number of samples in a signal be a power of 2 (e.g., 512, 1024, 2048 samples, etc.)

7/3/2015 Biomechanics Laborartory, University of Ottawa15 Fourier Transforms of Known Waveforms   Sine wave: w(t)=a sin(wt)   Square wave: w(t)=a [sin(  t) + 1/3 sin(3  t) + 1/5 sin(5  t) +... ]   Triangle wave: w(t)=8a/  2 [cos(  t) + 1/9 cos(3  t) + 1/25 cos(5  t) +...]   Sawtooth wave: w(t)=2a/  [sin(  t) – 1/2 sin(2  t) + 1/3 sin(3  t) – 1/4 sin(4  t) + 1/5 sin(5  t) +...]

7/3/2015 Biomechanics Laborartory, University of Ottawa16 Pezzack’s Angular Displacement Data

7/3/2015 Biomechanics Laborartory, University of Ottawa17 Fourier Analysis of Pezzack’s Angular Displacement Data Bias = a 0 = HarmonicFreq.c i b i Normalized number(hertz)cos(  )sin(  )power

7/3/2015 Biomechanics Laborartory, University of Ottawa18 Reconstruction of Pezzack’s Angular Displacement Data raw signal (green) 8 harmonics (cyan) 4 harmonics (red) 2 harmonics (magenta) 8 harmonics gave a reasonable approximation